In a right triangle, the tangent of an acute angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
Triangle:
Trigonometric Ratios:
sin θ=24/25,
cos θ=7/25,
csc θ=25/24,
sec θ=25/7,
cot θ=7/24
Practice makes perfect
Given that tan θ= 247, we want to sketch a right triangle with θ as the measure of one acute angle. Then, we will find the other five trigonometric ratios of θ. Let's do these things one at a time.
Drawing the Triangle
In a right triangle, the tangent of an acute angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
tan θ =24/7 ⇔ tan θ = opposite/adjacent
Therefore, we know that the length of the opposite side to θ is 24 and that the length of the adjacent side to θ is 7.
Note that when solving the equation we only considered the principal root. This is because c represents a side length and therefore must be a positive number. We can now draw the right triangle and label its three sides.
Finding Trigonometric Ratios
Having the three sides of the right triangle allows us to find the five remaining trigonometric ratios.
